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Combination

Evaluate 47...

Question

Evaluate
$^{47} C_{4} + 3 \sum j = 0^{50 - j} C_{3} + 5 \sum k = 0^{56 - k} C_{53 - k}$

A

^{57} C_{4}

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B

^{57} C_{5}

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C

^{56} C_{6}

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D

^{56} C_{5}

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Solution

The correct option is A $^{57} C_{4}$
Expanding , we get
$[^{47} C_{4} +^{47} C_{3} +^{48} C_{3} +^{49} C_{3} +^{50} C_{3}] + [^{56} C_{53} +^{55} C_{52} +^{54} C_{51} +^{53} C_{50} +^{52} C_{49} +^{51} C_{58}]$
Now
$^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}$
Hence
$[^{47} C_{4} +^{47} C_{3} +^{48} C_{3} +^{49} C_{3} +^{50} C_{3}$
$=^{48} C_{4} +^{48} C_{3} +^{49} C_{3} +^{50} C_{3}$
:
:
$=^{51} C_{4}$
Now
$^{56} C_{53} +^{55} C_{52} +^{54} C_{51} +^{53} C_{50} +^{52} C_{49} +^{51} C_{58}$
$=^{56} C_{3} +^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} +^{51} C_{3}$
$^{n} C_{r} =^{n} C_{n - r}$
Hence Adding we get
$[^{56} C_{3} +^{55} C_{3} +^{54} C_{3} +^{53} C_{3} +^{52} C_{3} +^{51} C_{3}] +^{51} C_{4}$
$=^{56} C_{3} +^{56} C_{4}$
$=^{57} C_{4}$

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